Plateau's Problem Frederick J., Jr. AlmgrenThere have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past 25 to 30 years. Many of the researchers who have produced these excellent results were inspired by this little book - or by Fred Almgren himself. The book is indeed a delightful invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behavior of soap films.When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This new tool makes possible much exciting new analysis and many new results. Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. The author's hope in writing this book was to encourage young mathematicians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well.Nonlinear Functional Analysis & Applications to Differential Equations Antonio Ambrosetti, K. -C Chang, Kwang-Chih ChangAn exploration of nonlinear functional analysis and applications to differential equations. It addresses areas such as Hamiltonian systems, variational methods, duality in nonlinear analysis and applications, and degree and multiplicity results for a superlinear problem.Functions of Bounded Variation and Free Discontinuity Problems Luigi Ambrosio, Nicola Fusco, Diego PallaraThis book deals with a class of mathematical problems which involve the minimization of the sum of a volume and a surface energy and have lately been referred to as 'free discontinuity problems'. The aim of this book is twofold: The first three chapters present all the basic prerequisites for the treatment of free discontinuity and other variational problems in a systematic, general, and self-contained way. In the later chapters, the reader is introduced to the theory of free discontinuity problems, to the space of special functions of bounded variation, and is presented with a detailed analysis of the Mumford-Shah image segmentation problem. Existence, regularity and qualitative properties of solutions are explained and a survey is given on the current knowledge of this challenging mathematical problem. The theory embodies classical problems, e.g. related to phase transitions, or fracture and plasticity in continuum mechanics, as well as more recent ones like edge detection in image analysis. This book provides the reader with a solid introduction to the field, written by principle contributors to the theory.102 Combinatorial Problems Titu Andreescu, Zuming Feng"102 Combinatorial Problems" consists of carefully selected problems that have been used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Key features: * Provides in-depth enrichment in the important areas of combinatorics by reorganizing and enhancing problem-solving tactics and strategies * Topics include: combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, theory of equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations and classical inequalities The book is systematically organized, gradually building combinatorial skills and techniques and broadening the student's view of mathematics. Aside from its practical use in training teachers and students engaged in mathematical competitions, it is a source of enrichment that is bound to stimulate interest in a variety of mathematical areas that are tangential to combinatorics.Number Theory G. AndrewsMathematical Analysis, Second Edition Tom M. ApostolAlgebraic Numbers and Algebraic Functions Emil ArtinFamous Norwegian mathematician Niels Henrik Abel advised that one should "learn from the masters, not from the pupils". When the subject is algebraic numbers and algebraic functions, there is no greater master than Emil Artin. In this classic text, originated from the notes of the course given at Princeton University in 1950-1951 and first published in 1967, one has a beautiful introduction to the subject accompanied by Artin's unique insights and perspectives. The exposition starts with the general theory of valuation fields in Part I, proceeds to the local class field theory in Part II, and then to the theory of function fields in one variable (including the Riemann-Roch theorem and its applications) in Part III. Prerequisites for reading the book are a standard first-year graduate course in algebra (including some Galois theory) and elementary notions of point set topology. With many examples, this book can be used by graduate students and all mathematicians learning number theory and related areas of algebraic geometry of curves.Algebra, Second Edition Michael ArtinGeometry of Yang-Mills fields Michael F. AtiyahThese Lecture Notes are an expanded version of the Fermi Lectures I gave at Scuola Normale Superiore in Pisa, the Loeb Lectures at Harvard and the Whittemore Lectures at Yale, in 1978. In all cases I was addressing a mixed audience of mathematicians and physicists and the presentation had to be tailored accordingly. Throughout, I presented the mathematical material in a somewhat unorthodox order, following a pattern which I felt would relate the new techniques to familiar ground for physicists. The main new results presented in the lectures, namely the construction of all multi-istanton solutions of Yang-Mills fields, is the culmination of several years of fruitful interaction between many physicists and mathematicians. The major breakthrough came with the observation by Ward that the complex methods developed by Penrose in his “twistor programme” were ideally suited to the study of the Yang-Mills equations. The instanton problem was then seen to be equivalent to a problem in complex analysis and to one in algebraic geometry. Using the powerful methods of modern algebraic geometry it was not long before the problem was finally solved.Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations Gilles Aubert, Pierre KornprobstThe updated 2nd edition of this book presents a variety of image analysis applications, reviews their precise mathematics and shows how to discretize them. For the mathematical community, the book shows the contribution of mathematics to this domain, and highlights unsolved theoretical questions. For the computer vision community, it presents a clear, self-contained and global overview of the mathematics involved in image procesing problems. The second edition offers a review of progress in image processing applications covered by the PDE framework, and updates the existing material. The book also provides programming tools for creating simulations with minimal effort.Transcendental Number Theory Alan BakerFirst published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979, however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.Creative Mathematics: A Gateway to Research Alan F. BeardonHow do mathematicians approach a problem, explore the possibilities, and develop an understanding of a whole area around it? The issue is not simply about obtaining 'the answer'; rather, Beardon explains that a mathematical problem is just one of many related ones that should be simultaneously investigated and discussed at various levels, and that understanding this is a crucial step in becoming a creative mathematician. The book begins with some good advice about procedure, presentation, and organization that will benefit every mathematician, budding, teaching or practiced. In the rest of the book, Beardon presents a series of simple problems, then, through discussion, consideration of special cases, computer experiments, and so on, the reader is taken through these same problems, but at an increasing level of sophistication and generality. Mathematics is rarely a closed book, and seemingly innocent problems, when examined and explored, can lead to results of significance.Topology Optimization Martin Philip Bendsoe, Ole SigmundThe topology optimization method solves the basic enginee- ring problem of distributing a limited amount of material in a design space. The first edition of this book has become the standard text on optimal design which is concerned with the optimization of structural topology, shape and material. This edition, has been substantially revised and updated to reflect progress made in modelling and computational procedures. It also encompasses a comprehensive and unified description of the state-of-the-art of the so-called material distribution method, based on the use of mathematical programming and finite elements. Applications treated include not only structures but also materials and MEMS.Geometry I Marcel BergerVolume I of this 2-volume textbook provides a lively and readable presentation of large parts of classical geometry. For each topic the author presents an esthetically pleasing and easily stated theorem - although the proof may be difficult and concealed. The mathematical text is illustrated with figures, open problems and references to modern literature, providing a unified reference to geometry in the full breadth of its subfields and ramifications.A Panoramic View of Riemannian Geometry Marcel BergerThis book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field.From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." —MATHEMATICAL REVIEWS Fourier Series Rajendra BhatiaDiscrete Mathematics Norman L. BiggsThe widespread use of computers and the rapid growth in computer science have led to a new emphasis on discrete mathematics, a discipline which deals with calculations involving a finite number of steps. This book provides a well-structured introduction to discrete mathematics, taking a self-contained approach that requires no ancillary knowledge of mathematics, avoids unnecessary abstraction, and incorporates a wide rage of topics, including graph theory, combinatorics, number theory, coding theory, combinatorial optimization, and abstract algebra. Amply illustrated with examples and exercises.Probability and Measure Patrick BillingsleyPROBABILITY AND MEASUREThird Edition Now in its new third edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure, so that probability problems generate an interest in measure theory and measure theory is then developed and applied to probability. Probability and Measure provides thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. The Third Edition features an improved treatment of Brownian motion and the replacement of queuing theory with ergodic theory. Like the previous editions, this new edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory. Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein, Adrian S. LewisOptimization is a rich and thriving mathematical discipline, and the underlying theory of current computational optimization techniques grows ever more sophisticated. This book aims to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. Each section concludes with an often extensive set of optional exercises. This new edition adds material on semismooth optimization, as well as several new proofs.Exploring the Number Jungle: A Journey into Diophantine Analysis Edward B. BurgerWelcome to diophantine analysis—an area of number theory in which we attempt to discover hidden treasures and truths within the jungle of numbers by exploring rational numbers. Diophantine analysis comprises two different but interconnected domains—diophantine approximation and diophantine equations. This highly readable book brings to life the fundamental ideas and theorems from diophantine approximation, geometry of numbers, diophantine geometry and $p$-adic analysis. Through an engaging style, readers participate in a journey through these areas of number theory. Each mathematical theme is presented in a self-contained manner and is motivated by very basic notions. The reader becomes an active participant in the explorations, as each module includes a sequence of numbered questions to be answered and statements to be verified. Many hints and remarks are provided to be freely used and enjoyed. Each module then closes with a Big Picture Question that invites the reader to step back from all the technical details and take a panoramic view of how the ideas at hand fit into the larger mathematical landscape. This book enlists the reader to build intuition, develop ideas and prove results in a very user-friendly and enjoyable environment.Fully Nonlinear Elliptic Equations (Colloquium Publications (Amer Mathematical Soc)) Luis A. Caffarelli, Xavier CabreThis book provides a self-contained development of the regularity theory for solutions of fully nonlinear elliptic equations. Caffarelli and Cabré offer a detailed presentation of all techniques needed to extend the classical Schauder and Calderón-Zygmund regularity theories for linear elliptic equations to the fully nonlinear context.The authors present the key ideas and prove all the results needed for the regularity theory of viscosity solutions of fully nonlinear equations. The book contains the study of convex fully nonlinear equations and fully nonlinear equations with variable coefficients. Calculus of Variations and Partial Differential Equations of First Order Constantin Caratheodory, C. CarathéodoryIn this second English edition of Carathéodory's famous work (originally published in German), the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the calculus of variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Carathéodory's masterpiece. Includes a Guide to the Literature and an Index. From the Preface: "The book consists of two parts. In the first part, I have made an attempt to simplify the presentation of the theory of partial differential equations to the first order so that its study will require little time and also be accessible to the average student of mathematics ... The second part, which contains the Calculus of Variations, can also be read independently if one refers back to earlier sections in Part I ... I have never lost sight of the fact that the Calculus of Variations, as it is presented in Part II, should above all be a servant of Mechanics. Therefore, I have in particular prepared everything from the very outset for treatment in multidimensional spaces."Methods in Nonlinear Analysis Kung-Ching ChangThis book offers a systematic presentation of up-to-date material scattered throughout the literature from the methodology point of view. It reviews the basic theories and methods, with many interesting problems in partial and ordinary differential equations, differential geometry and mathematical physics as applications, and provides the necessary preparation for almost all important aspects in contemporary studies. All methods are illustrated by carefully chosen examples from mechanics, physics, engineering and geometry.Riemannian Geometry: A Modern Introduction Isaac ChavelRequiring only an understanding of differentiable manifolds, Isaac Chavel covers introductory ideas followed by a selection of more specialized topics in this second edition. He provides a clearer treatment of many topics, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. Among the classical topics shown in a new setting is isoperimetric inequalities in curved spaces. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.Hamilton's Ricci Flow Bennett Chow, Peng Lu, and Lei NiRicci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the PoincarÃ© conjecture and Thurston's geometrization conjecture.A Course in Functional Analysis John B ConwayThis book is an introductory text in functional analysis. Unlike many modern treatments, it begins with the particular and works its way to the more general.From the reviews: "This book is an excellent text for a first graduate course in functional analysis....Many interesting and important applications are included....It includes an abundance of exercises, and is written in the engaging and lucid style which we have come to expect from the author." —MATHEMATICAL REVIEWS Weakly Nonlinear Dirichlet Problems on Long or Thin Domains E. N. DancerThe aim of this work is to develop a basic theory for nonlinear elliptic equations on long or thin domains for Dirichlet boundary conditions. This is the first treatment of such Dirichlet problems, which are of significant interest in applications.Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5 Evolution Problems I Robert Dautray, Jacques-Louis LionsThese 6 volumes - the result of a 10 year collaboration between the authors, two of France's leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences.Heat Kernels and Spectral Theory E. B. DaviesAn advanced monograph on a central topic in the theory of differential equations, Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators. While the study of the heat equation is a classical subject, this book analyses the improvements in our quantitative understanding of heat kernels. The author considers variable coefficient operators on regions in Euclidean space and Laplace-Beltrami operators on complete Riemannian manifolds. He also includes results pertaining to the heat kernels of Schrödinger operators; such results will be of particular interest to mathematical physicists, and relevant too to those concerned with properties of Brownian motion and other diffusion processes.Nonlinear Functional Analysis Klaus Deimling, MathematicsHailed as "eminently suitable as a text for a graduate course" by the Bulletin of the American Mathematical Society, this volume offers a survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis. It offers extensive commentary and many examples in addition to an abundance of interesting, challenging exercises.Starting with coverage of the development of the Brower degree and its applications, the text proceeds to examinations of degree mappings for infinite dimensional spaces and surveys of monotone and accretive mappings. Subsequent chapters explore the inverse function theory, the implicit function theory, and Newton's methods as well as fixed-point theory, solutions to cones, and the Galerkin method of studying nonlinear equations. The final chapters address extremal problems—including convexity, Lagrange multipliers, and mini-max theorems—and offer an introduction into bifurcation theory. Suitable for graduate-level mathematics courses, this volume also serves as a reference for professionals. History of the Theory of Numbers , Vol. 2: Diophantine Analysis Leonard Eugene DicksonThis second volume is a comprehensive treatment of Diophantine analysis. Besides the familiar cases of Diophantine equations, this rubric also covers partitions, representations as a sum of two, three, four or $n$ squares, Waring's problem in general and Hilbert's solution of it, and perfect squares in artihmetical and geometrical progressions. Of course, many important Diophantine equations, such as Pell's equation, and classes of equations, such as quadratic, cubic and quartic equations, are treated in detail. As usual with Dickson, the account is encyclopedic and the references are numerous.Lectures on Number Theory Peter Gustav Lejeune Dirichlet, Richard Dedekind, P. G. L. DirichletThis volume is a translation of Dirichlet's Vorlesungen über Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume.Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions. The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory. The legendary story is told how Dirichlet kept a copy of Gauss's Disquisitiones Arithmeticae with him at all times and how Dirichlet strove to clarify and simplify Gauss's results. Dedekind's footnotes document what material Dirichlet took from Gauss, allowing insight into how Dirichlet transformed the ideas into essentially modern form. Also shown is how Gauss built on a long tradition in number theory—going back to Diophantus—and how it set the agenda for Dirichlet's work. This important book combines historical perspective with transcendent mathematical insight. The material is still fresh and presented in a very readable fashion. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. (For another historical translation by Professor Stillwell, see Sources of Hyperbolic Geometry, Volume 10 in the History of Mathematics series.) Topology James DugundjiConvexity H. G. EgglestonThis account of convexity includes the basic properties of convex sets in Euclidean space and their applications, the theory of convex functions and an outline of the results of transformations and combinations of convex sets. It will be useful for those concerned with the many applications of convexity in economics, the theory of games, the theory of functions, topology, geometry and the theory of numbers.Convexity Methods in Hamiltonian Mechanics IVAR EKELANDDifferential Equations Methods for the Monge-Kantorevich Mass Transfer Problem Lawrence C. Evans, Wilfrid GangboIn this volume, the authors demonstrate under some assumptions on $f^+$, $f^-$ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{^+}=f^+dx$ onto $\mu^-=f^-dy$ can be constructed by studying the $p$-Laplacian equation $- \mathrm {div}(\vert DU_p\vert^{p-2}Du_p)=f^+-f^-$ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1,-\mathrm {div}(aDu)=f^+-f^-$ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f^+$ and $f^-$.Partial Differential Equations Lawrence C. EvansThis text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between functional analytic insights and calculus-type estimates within the context of Sobolev spaces. Treatment of all topics is complete and self-contained. The book's wide scope and clear exposition make it a suitable text for a graduate course in PDEs.Weak Convergence Methods for Nonlinear Partial Differential Equations (Regional Conference Seriess in Mathematics, No 74) CBMS/74 (Cbms Regional Conference Series in Mathematics) Lawrence C. Evanspurpose of this book is to explain systematically and clearly many of the most important techniques set forth in recent years for using weak convergence methods to study nonlinear partial differential equations. This work represents an expanded version of a series of ten talks presented by the author at Loyola University of Chicago in the summer of 1988. The author surveys a wide collection of techniques for showing the existence of solutions to various nonlinear partial differential equations, especially when strong analytic estimates are unavailable. The overall guiding viewpoint is that when a sequence of approximate solutions converges only weakly, one must exploit the nonlinear structure of the PDE to justify passing to limits. The author concentrates on several areas that are rapidly developing and points to some underlying viewpoints common to them all. Among the several themes in the book are the primary role of measure theory and real analysis (as opposed to functional analysis) and the continual use in diverse settings of low amplitude, high frequency periodic test functions to extract useful information. The author uses the simplest problems possible to illustrate various key techniques. Aimed at research mathematicians in the field of nonlinear PDEs, this book should prove an important resource for understanding the techniques being used at the forefront of this vital area of researchGroups and Symmetry: A Guide to Discovering Mathematics David W. FarmerIn most mathematics textbooks, the most exciting part of mathematics—the process of invention and discovery—is completely hidden from the reader. The aim of Groups and Symmetry is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: Its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you.Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement. Knots and Surfaces: A Guide to Discovering Mathematics David W. Farmer, Theodore B. StanfordIn most mathematics textbooks, the most exciting part of mathematics—the process of invention and discovery—is completely hidden from the reader. The aim of Knots and Surfaces is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you.Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement. Fourier Analysis and Its Applications Gerald B. FollandThis book presents the theory and applications of Fourier series and integrals, eigenfunction expansions, and related topics, on a level suitable for advanced undergraduates. It includes material on Bessel functions, orthogonal polynomials, and Laplace transforms, and it concludes with chapters on generalized functions and Green's functions for ordinary and partial differential equations. The book deals almost exclusively with aspects of these subjects that are useful in physics and engineering, and includes a wide variety of applications. On the theoretical side, it uses ideas from modern analysis to develop the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs.A First Course in Abstract Algebra Seventh Edition John B. FraleighTopics in the Calculus of Variations MARTIN FUCHSSome Points in Analysis and Their History (University Lecture Series, Vol 11) ULECT/11 Lars GardingThis book is a collection of small essays containing the history and the proofs of some important and interesting theorems of analysis and partial differential operators in this century. Most of the results in the book are associated with Swedish mathematicians. Also included are the Tarski-Seidenberg theorem and Wiener's classical results in harmonic analysis and a delightful essay on the impact of distributions in analysis. All mathematical points are fully explained, but some require a certain mature understanding from the reader. This book is a well-written, simple work that offers full mathematical treatment, along with insight and fresh points of view.Geometric Tomography Richard J. GardnerGeometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This edition, first published in 2006, includes numerous updates and improvements, with some 300 new references bringing the total to over 800.Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Mariano GiaquintaBook by Giaquinta, MarianoElliptic Partial Differential Equations of Second Order David Gilbarg, Neil S. TrudingerFrom the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student." —New Zealand Mathematical Society, 1985Introduction to Algebraic Curves Phillip A. GriffithsAlgebraic curves and compact Riemann surfaces comprise the most developed and arguably the most beautiful portion of algebraic geometry. However, the majority of books written on the subject discuss algebraic curves and compact Riemann surfaces separately, as parts of distinct general theories. Most texts and university courses on curve theory generally conclude with the Riemann-Roch theorem, despite the fact that this theorem is the gateway to some of the most fascinating results in the theory of algebraic curves. This book is based on a six-week series of lectures presented by the author to third- and fourth-year undergraduates and graduate students at Beijing University in 1982. The lectures began with minimal technical requirements (a working knowledge of elementary complex function theory and algebra together with some exposure to topology of compact surfaces) and proceeded directly to the Riemann-Roch and Abel theorems. This book differs from a number of recent books on this subject in that it combines analytic and geometric methods at the outset, so that the reader can grasp the basic results of the subject. Although such modern techniques of sheaf theory, cohomology, and commutative algebra are not covered here, the book provides a solid foundation to proceed to more advanced texts in general algebraic geometry, complex manifolds, and Riemann surfaces, as well as algebraic curves. Containing numerous exercises and two exams, this book would make an excellent introductory text.Elliptic Partial Differential Equations: Second Edition Qing Han, Fanghua LinElliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things about it—it is a wonderful book. —Tobias Colding This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems. This second edition has been thoroughly revised and in a new chapter the authors discuss several methods for proving the existence of solutions of primarily the Dirichlet problem for various types of elliptic equations. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.An Introduction to the Theory of Numbers G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph SilvermanAn Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.Updates include a chapter by J. H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem — a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader. The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists. Ordinary Differential Equations Philip HartmanOrdinary Differential Equations covers the fundamentals of the theory of ordinary differential equations (ODEs), including an extensive discussion of the integration of differential inequalities, on which this theory relies heavily. In addition to these results, the text illustrates techniques involving simple topological arguments, fixed point theorems, and basic facts of functional analysis. Unlike many texts, which supply only the standard simplified theorems, Ordinary Differential Equations presents the basic theory of ODEs in a general way, making it a valuable reference. This SIAM reissue of the 1982 second edition covers invariant manifolds, perturbations, and dichotomies, making the text relevant to current studies of geometrical theory of differential equations and dynamical systems.Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities Emmanuel HebeyThis volume offers an expanded version of lectures given at the Courant Institute on the theory of Sobolev spaces on Riemannian manifolds. "Several surprising phenomena appear when studying Sobolev spaces on manifolds," according to the author. "Questions that are elementary for Euclidean space become challenging and give rise to sophisticated mathematics, where the geometry of the manifold plays a central role." The volume is organized into nine chapters. Chapter 1 offers a brief introduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces for compact manifolds. Chapter 3 presents the general theory of Sobolev spaces for complete, noncompact manifolds. Best constants problems for compact manifolds are discussed in Chapters 4 and 5. Chapter 6 presents special types of Sobolev inequalities under constraints. Best constants problems for complete noncompact manifolds are discussed in Chapter 7. Chapter 8 deals with Euclidean-type Sobolev inequalities. And Chapter 9 discusses the influence of symmetries on Sobolev embeddings. An appendix offers brief notes on the case of manifolds with boundaries. This topic is a field undergoing great development at this time. However, several important questions remain open. So a substantial part of the book is devoted to the concept of best constants, which appeared to be crucial for solving limiting cases of some classes of PDEs. The volume is highly self-contained. No familiarity is assumed with differentiable manifolds and Riemannian geometry, making the book accessible to a broad audience of readers, including graduate students and researchers.Extremum Problems for Eigenvalues of Elliptic Operators Antoine HenrotThis book focuses on extremal problems. For instance, it seeks a domain which minimizes or maximizes a given eigenvalue of the Laplace operator with various boundary conditions and various geometric constraints. Also considered is the case of functions of eigenvalues. The text probes similar questions for other elliptic operators, such as Schrodinger, and explores optimal composites and optimal insulation problems in terms of eigenvalues.Introduction to Stochastic Processes Paul G/ Port, Sidney C/ Stone, Charles J HoelFudamentals of Banach algebras K. HoffmanThe Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis Lars HörmanderThe main change in this edition is the inclusion of exercises with answers and hints. This is meant to emphasize that this volume has been written as a general course in modern analysis on a graduate student level and not only as the beginning of a specialized course in partial differen tial equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of gen eral interest; there are none to the last two chapters. Most of the exercises are just routine problems meant to give some familiarity with standard use of the tools introduced in the text. Others are extensions of the theory presented there. As a rule rather complete though brief solutions are then given in the answers and hints. To a large extent the exercises have been taken over from courses or examinations given by Anders Melin or myself at the University of Lund. I am grateful to Anders Melin for letting me use the problems originating from him and for numerous valuable comments on this collection. As in the revised printing of Volume II, a number of minor flaws have also been corrected in this edition. Many of these have been called to my attention by the Russian translators of the first edition, and I wish to thank them for our excellent collaboration.Mechanical Behavior of Materials William F. HosfordThis textbook is for courses on Mechanical Behavior of Materials taught in departments of Mechanical Engineering and Materials Science. The text includes numerous examples and problems for student practice. The book emphasizes quantitative problem solving. End of the chapter notes are included to increase students' interest. This text differs from others because the treatment of plasticity has greater emphasis on the interrelationship of the flow, effective strain and effective stress and their use in conjunction with yield criteria to solve problems. The treatment of defects is new. Schmid's law is generalized for complex stress states. Its use with strains allows for prediction of R-values for textures. Another feature is the treatment of lattice rotations and how they lead to deformation textures. The chapter on fracture mechanics includes coverage of Gurney's approach. Much of the analysis of particulate composites is new. Few texts include anything on metal forming.Inverse Problems for Partial Differential Equations VICTOR ISAKOVBasic Algebra I: Second Edition Nathan JacobsonPlane Waves and Spherical Means Applied to Partial Differential Equations Fritz JohnAbroad mathematical classics series: Numerical Optimization Wright Stephen J Nocedal JorgeAbstracts. International Workshop on Stochastic Filtering Theory, Held in Chapel Hill, North Carolina on June 26-28, 1994 G. KallianpurBrownian Motion and Stochastic Calculus Ioannis Karatzas, Steven ShreveA graduate-course text, written for readers familiar with measure-theoretic probability and discrete-time processes, wishing to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed, illustrated by results concerning representations of martingales and change of measure on Wiener space, which in turn permit a presentation of recent advances in financial economics. The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The whole is backed by a large number of problems and exercises.P-adic Analysis Compared With Real Svetlana KatokThe book gives an introduction to $p$-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity of isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and $p$-adic contexts of the book. The book is based on an advanced undergraduate course given by the author. The choice of the topic was motivated by the internal beauty of the subject of $p$-adic analysis, an unusual one in the undergraduate curriculum, and abundant opportunities to compare it with its much more familiar real counterpart. The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. Well written, with obvious care for the reader, the book can be successfully used in a topic course or for self-study.Geometric Inequalities Nicholas D. KazarinoffAnybody who liked his first geometry course will enjoy the simply stated geometric problems about maximum and minimum lenghs and areas in this book. Many of these already fascinated the greeks, for example the problem of of enclosing the largest possible area by a fence of given length, and some were solved long ago; but others remain unsolved even today. Some of the solutions of the problems posed in this book, for example the problem of inscribing a triangle of smallest perimeter into a given triangle, were supplied by world famous mathemaicians, others by high school students.Prescribing the Curvature of a Riemannian Manifold Jerry L. KazdanThese notes were the basis for a series of ten lectures given in January 1984 at Polytechnic Institute of New York under the sponsorship of the Conference Board of the Mathematical Sciences and the National Science Foundation. The lectures were aimed at mathematicians who knew either some differential geometry or partial differential equations, although others could understand the lectures. Author's Summary:Given a Riemannian Manifold $(M,g)$ one can compute the sectional, Ricci, and scalar curvatures. In other special circumstances one also has mean curvatures, holomorphic curvatures, etc. The inverse problem is, given a candidate for some curvature, to determine if there is some metric $g$ with that as its curvature. One may also restrict ones attention to a special class of metrics, such as Kahler or conformal metrics, or those coming from an embedding. These problems lead one to (try to) solve nonlinear partial differential equations. However, there may be topological or analytic obstructions to solving these equations. A discussion of these problems thus requires a balanced understanding between various existence and non-existence results. The intent of this volume is to give an up-to-date survey of these questions, including enough background, so that the current research literature is accessible to mathematicians who are not necessarily experts in PDE or differential geometry. The intended audience is mathematicians and graduate students who know either PDE or differential geometry at roughly the level of an intermediate graduate course.Functional Analysis S. KesavanThe material presented in this book is suited for a first course in Functional Analysis which can be followed by Masters students. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of various theorems via examples taken from differential equations and the calculus of variations, either through brief sections or through exercises. In fact, this book will be particularly useful for students who would like to pursue a research career in the applications of mathematics. The book includes a chapter on weak and weak topologies and their applications to the notions of reflexivity, separability and uniform convexity. The chapter on the Lebesgue spaces also presents the theory of one of the simplest classes of Sobolev spaces. The book includes a chapter on compact operators and the spectral theory for compact self-adjoint operators on a Hilbert space. Each chapter has large collection of exercises at the end. These illustrate the results of the text, show the optimality of the hypotheses of various theorems via examples or counterexamples, or develop simple versions of theories not elaborated upon in the text.An Introduction to Variational Inequalities and Their Applications David Kinderlehrer, Guido StampacchiaThis unabridged republication of the 1980 text, an established classic in the field, is a resource for many important topics in elliptic equations and systems and is the first modern treatment of free boundary problems. Variational inequalities (equilibrium or evolution problems typically with convex constraints) are carefully explained in An Introduction to Variational Inequalities and Their Applications. They are shown to be extremely useful across a wide variety of subjects, ranging from linear programming to free boundary problems in partial differential equations. Exciting new areas like finance and phase transformations along with more historical ones like contact problems have begun to rely on variational inequalities, making this book a necessity once again.Introduction to the Theory of Diffusion Processes Nikolai Vladimirovich KrylovFocusing on one of the major branches of probability theory, this book treats the large class of processes with continuous sample paths that possess the ``Markov property''. The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of Ito's stochastic integral equations. The book is designed as a self-contained introduction, requiring no background in the theory of probability or even in measure theory. In particular, the theory of local continuous martingales is covered without the introduction of the idea of conditional expectation. Krylov covers such subjects as the Wiener process and its properties, the theory of stochastic integrals, stochastic differential equations and their relation to elliptic and parabolic partial differential equations, Kolmogorov's equations, and methods for proving the smoothness of probabilistic solutions of partial differential equations. With many exercises and thought-provoking problems, this book would be an excellent text for a graduate course in diffusion processes and related subjects.Linear Algebra K. Hoffman, R. KunzeMathematical Pamphlets Algebraic Number Theory R. Narasimhan, S. Raghavan, S.S. Rangachari, Sunder LalHyperbolic Partial Differential Equations Peter D. LaxThe theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of solutions of equations with constant coefficients is explored with the help of the Fourier and Radon transforms. The existence of solutions of equations with variable coefficients with prescribed initial values is proved using energy inequalities. The propagation of singularities is studied with the help of progressing waves. The second part describes finite difference approximations of hyperbolic equations, presents a streamlined version of the Lax-Phillips scattering theory, and covers basic concepts and results for hyperbolic systems of conservation laws, an active research area today. Four brief appendices sketch topics that are important or amusing, such as Huygens' principle and a theory of mixed initial and boundary value problems. A fifth appendix by Cathleen Morawetz describes a nonstandard energy identity and its uses. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.Geometric Measure Theory: An Introduction Fanghua Lin, Xiaoping YangSince the publication of the seminal work of H. Federer which gives a rather complete and comprehensive discussion on the subject, the geometric measure theory has developed in the last three decades into an even more cohesive body of basic knowledge with an ample structure of its own, establishing strong ties with many other areas of mathematics and made numerous new striking applications. The present book is intended for the researchers in other fields of mathematics as well as graduate students for a quick overview on the subject of the geometric measure theory emphasizing on various basic ideas, techniques and their applications in problems arising in the calculus of variations, geometrical analysis and nonlinear partial differential equations. This graduate-level treatment of Geometric Measure Theory illustrates with concrete examples and emphasizes basic ideas and techniques with their applications to the calculus of variations, geometrical analysis, and nonlinear PDEs. The book, in addition to a full index and bibliography, include eight main chapters. New paperback re-issue.Analysis E.H. Lieb, M. LossElements of Functional Analysis L.A.; Sobolev, V.J. LusternikMorse Theory John MilnorOne of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory.Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master." Milnor has published five books with Princeton University Press. The Theory of Composites Graeme W. MiltonThe theory of composite materials is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively studied for more than a hundred years, an explosion of ideas in the past four decades has dramatically increased our understanding of the relationship among the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli that govern the macroscopic behavior. This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification. This book surveys these exciting developments at the frontier of mathematics and presents many new results.Probabilità ed informazione A.C.G. Mennucci, S.K. MitterLectures on Kähler Geometry Andrei MoroianuKähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained 2007 graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi-Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.The Red Book of Varieties and Schemes. Includes the Michigan Lectures (1974) on Curves and their Jacobians DAVID MUMFORDTopology James MunkresOrlicz Spaces and Modular Spaces J. MusielakRepresentations of Finite Groups C. MusiliTopics in nonlinear functional analysis, 1973-1974 L NirenbergExcursions in Geometry C. Stanley OgilvyTopics including harmonic division and Apollonian circles, inversive geometry, the hexlet, conic sections, projective geometry, the Golden Section and angle trisection are addressed in a way that brings out the true intellectual excitement inherent in each. Also included: some unsolved problems of modern geometry. Notes. References. 132 line illustrations.Stochastic Differential Equations an Introduction with Applications Sixth Edition Bernt OksendalThis Book is Brand new international softcover edition delivered within 7-12 working days via UPS/USPS/DHL and FEDEX.(FOR SALE ONLY U.S. & U.K.)Introduction to Complex Analysis R. Nevanlinna, V. PaateroIntroduction to Probability and Measure K.R. ParthasarathyHow to Solve it G. PolyaIsoperimetric Inequalities in Mathematical Physics George PolyaLectures in Geometry, Analytic Geometry M. PostnikovLectures in Geometry, Lie groups and Lie algebras M. PostnikovLectures in Geometry, Smooth manifolds M. PostnikovMaximum Principles in Differential Equations Murray H. Protter, Hans F. WeinbergerMaximum Principles are central to the theory and applications of second-order partial differential equations and systems. This self-contained text establishes the fundamental principles and provides a variety of applications.Lectures on Topics in one-parameter bifurcation problems P. RabierMinimax Methods in Critical Point Theory with Applications to Differential Equations Paul H. RabinowitzThe book provides an introduction to minimax methods in critical point theory and shows their use in existence questions for nonlinear differential equations. An expanded version of the author's 1984 CBMS lectures, this volume is the first monograph devoted solely to these topics. Among the abstract questions considered are the following: the mountain pass and saddle point theorems, multiple critical points for functionals invariant under a group of symmetries, perturbations from symmetry, and variational methods in bifurcation theory. The book requires some background in functional analysis and differential equations, especially elliptic partial differential equations. It is addressed to mathematicians interested in differential equations and/or nonlinear functional analysis, particularly critical point theory.An Introduction to Measure and Integration Inder K. RanaIntegration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, $L_p$ spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on. The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study. For this edition, more exercises and four appendices have been added.New Directions in Solid State Chemistry C. N. R. Rao, J. GopalakrishnanIn the new edition of this widely praised textbook the authors have brought the work completely up to date by the addition of new material on numerous topics, and all of the chapters have been revised. In recent years, solid state chemistry has become a vital component of mainstream chemistry and materials science. Students, teachers and researchers need to understand the chemistry of solids because of the crucial role this plays in determining the properties of materials. An understanding of solid state chemistry is also essential in materials design, and chemists continue to discover many fascinating relationships between the structure and properties of solids. This text requires only an understanding of basic physics, chemistry and crystallography, and contains the most recent examples, case studies and references. It will be a valuable reference for advanced students and researchers studying solid state chemistry and materials science.Hilbert Courant Constance ReidFunctional Analysis Frigyes Riesz, Bela Sz.-NagyClassic exposition of modern theories of differentiation and integration and principal problems and methods of handling integral equations and linear functionals and transformations. 1955 edition.Ordinary Differential Equations Garrett Birkhoff, Gian-Carlo RotaFunctional Analysis Walter RudinGeometric Partial Differential Equations and Image Analysis Guillermo SapiroThis book provides an introduction to the use of geometric partial differential equations in image processing and computer vision. It brings a number of new concepts into the field, providing a very fundamental and formal approach to image processing. State-of-the-art practical results in a large number of real problems are achieved with the techniques described. Applications covered include image segmentation, shape analysis, image enhancement, and tracking. The volume provides information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions.Algebraic Topology: An Intuitive Approach Hajime SatoThe single most difficult thing one faces when one begins to learn a new branch of mathematics is to get a feel for the mathematical sense of the subject. The purpose of this book is to help the aspiring reader acquire this essential common sense about algebraic topology in a short period of time. To this end, Sato leads the reader through simple but meaningful examples in concrete terms. Moreover, results are not discussed in their greatest possible generality, but in terms of the simplest and most essential cases. In response to suggestions from readers of the original edition of this book, Sato has added an appendix of useful definitions and results on sets, general topology, groups and such. He has also provided references.Topics covered include fundamental notions such as homeomorphisms, homotopy equivalence, fundamental groups and higher homotopy groups, homology and cohomology, fiber bundles, spectral sequences and characteristic classes. Objects and examples considered in the text include the torus, the Mobius strip, the Klein bottle, closed surfaces, cell complexes and vector bundles.Mathematical Pamphlets. Riemann Surfaces M.S. Narasimhan, R.R. Simha, R. Narasimhan, S. SeshadriMonotone Operators in Banach Space and Nonlinear Partial Differential Equations R. E. ShowalterThe objectives of this monograph are to present some topics from the theory of monotone operators and nonlinear semigroup theory which are directly applicable to the existence and uniqueness theory of initial-boundary-value problems for partial differential equations and to construct such operators as realizations of those problems in appropriate function spaces. A highlight of this presentation is the large number and variety of examples introduced to illustrate the connection between the theory of nonlinear operators and partial differential equations. These include primarily semilinear or quasilinear equations of elliptic or of parabolic type, degenerate cases with change of type, related systems and variational inequalities, and spatial boundary conditions of the usual Dirichlet, Neumann, Robin or dynamic type. The discussions of evolution equations include the usual initial-value problems as well as periodic or more general nonlocal constraints, history-value problems, those which may change type due to a possibly vanishing coefficient of the time derivative, and other implicit evolution equations or systems including hysteresis models. The scalar conservation law and semilinear wave equations are briefly mentioned, and hyperbolic systems arising from vibrations of elastic-plastic rods are developed. The origins of a representative sample of such problems is given in the Appendix.Differential equations with Applications and Historical Notes G.F. SimmonsTensor Calculus Barry SpainIntroduction to Riemann Surfaces George SpringerThis well-known book is a self-contained treatment of the classical theory of abstract Riemann surfaces. The first five chapters cover the requisite function theory and topology for Riemann surfaces. The second five chapters cover differentials and uniformization. For compact Riemann surfaces, there are clear treatments of divisors, Weierstrass points, the Riemann-Roch theorem and other important topics. Springer's book is an excellent text for an introductory course on Riemann surfaces. It includes exercises after each chapter and is illustrated with a beautiful set of figures.Mathematical Pamphlets Galois Theory M. Pavaman Murthy, K.G. Ramanathan, C.S. Seshadri, U. Shukla, R. SridharanHarmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Elias M. SteinThis book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L\sup\ estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems ,3/e Michael StruweMeasure and Probability S.R. Athreya And V.S. SunderDirect and Inverse Problems Potentials in Quantum Scattering B.N. Zakhariev, A.A. SuzkoThe theory of functions of a complex variable A. and Tikhonov, A. N. SveshnikovAn Introduction to Sobolev Spaces and Interpolation Spaces Luc TartarAfter publishing an introduction to the Navier–Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with another set of lecture notes based on a graduate course in two parts, as indicated by the title. A draft has been available on the internet for a few years. The author has now revised and polished it into a text accessible to a larger audience.Differential Geometry: Bundles, Connections, Metrics and Curvature Clifford Henry TaubesBundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kahler geometry. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.Mathematical Modeling in Continuum Mechanics Roger Temam, Alain MiranvilleTemam and Miranville present core topics within the general themes of fluid and solid mechanics. The brisk style allows the text to cover a wide range of topics including viscous flow, magnetohydrodynamics, atmospheric flows, shock equations, turbulence, nonlinear solid mechanics, solitons, and the nonlinear Schrödinger equation. This second edition will be a unique resource for those studying continuum mechanics at the advanced undergraduate and beginning graduate level whether in engineering, mathematics, physics or the applied sciences. Exercises and hints for solutions have been added to the majority of chapters, and the final part on solid mechanics has been substantially expanded. These additions have now made it appropriate for use as a textbook, but it also remains an ideal reference book for students and anyone interested in continuum mechanics.Selected Topics on "Analysis in Metric Spaces" L. Ambrosio, P. TilliIntroduction to the theory of Many-Body Systems A. Minguzzi, M. TosiModern Applied Statistics with S-PLUS W.N. Venables, B.D. RipleyS is a powerful environment for the statistical and graphical analysis of data. It provides the tools to implement many statistical ideas that have been made possible by the widespread availability of workstations having good graphics and computational capabilities. This book is a guide to using S environments to perform statistical analyses and provides both an introduction to the use of S and a course in modern statistical methods. Implementations of S are available commercially in S-PLUS(R) workstations and as the Open Source R for a wide range of computer systems. The aim of this book is to show how to use S as a powerful and graphical data analysis system. Readers are assumed to have a basic grounding in statistics, and so the book is intended for would-be users of S-PLUS or R and both students and researchers using statistics. Throughout, the emphasis is on presenting practical problems and full analyses of real data sets. Many of the methods discussed are state of the art approaches to topics such as linear, nonlinear and smooth regression models, tree-based methods, multivariate analysis, pattern recognition, survival analysis, time series and spatial statistics. Throughout modern techniques such as robust methods, non-parametric smoothing and bootstrapping are used where appropriate. This fourth edition is intended for users of S-PLUS 6.0 or R 1.5.0 or later. A substantial change from the third edition is updating for the current versions of S-PLUS and adding coverage of R. The introductory material has been rewritten to emphasis the import, export and manipulation of data. Increased computational power allows even more computer-intensive methods to be used, and methods such as GLMMs, MARS, SOM and support vector machines are considered.Topics in Optimal Transportation Cedric VillaniThis is the first comprehensive introduction to the theory of mass transportation with its many—and sometimes unexpected—applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook.In 1781, Gaspard Monge defined the problem of "optimal transportation" (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis. Generalized functions in Mathematical Physics V.S. VladimirovPercolation Et Modele D'ising / Percolation and the Ising Model Wendelin WernerThe Fourier Integral and Certain of its Applications Norbert WienerLectures on Harmonic Analysis Thomas H. Wolff, Izabella Aba, Carol ShubinThomas H. Wolff was a leading analyst and winner of the Salem and Bôcher Prizes. He made significant contributions to several areas of harmonic analysis, in particular to geometrical and measure-theoretic questions related to the Kakeya needle problem. Wolff attacked the problem with awesome power and originality, using both geometric and combinatorial ideas. This book provides an inside look at the techniques used and developed by Wolff. It is based on a graduate course on Fourier analysis he taught at Caltech. The selection of the material is somewhat unconventional in that it leads the reader, in Wolff's unique and straightforward way, through the basics directly to current research topics. The book demonstrates how harmonic analysis can provide penetrating insights into deep aspects of modern analysis. It is an introduction to the subject as a whole and an overview of those branches of harmonic analysis that are relevant to the Kakeya conjecture. The first few chapters cover the usual background material: the Fourier transform, convolution, the inversion theorem, the uncertainty principle, and the method of stationary phase. However, the choice of topics is highly selective, with emphasis on those frequently used in research inspired by the problems discussed in later chapters. These include questions related to the restriction conjecture and the Kakeya conjecture, distance sets, and Fourier transforms of singular measures. These problems are diverse, but often interconnected; they all combine sophisticated Fourier analysis with intriguing links to other areas of mathematics, and they continue to stimulate first-rate work. The book focuses on laying out a solid foundation for further reading and research. Technicalities are kept to a minimum, and simpler but more basic methods are often favored over the most recent methods. The clear style of the exposition and the quick progression from fundamentals to advanced topics ensure that both graduate students and research mathematicians will benefit from the book.Scattering Theory: Some Old and New Problems (Lecture Notes in Mathematics) Dmitri R. YafaevFunctional Analysis Kosaku YosidaThe present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i. e. , the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern and classical analysis. Necessary prerequisites for the reading of this book are summarized, with or without proof, in Chapter 0 under titles: Set Theory, Topo- logical Spaces, Measure Spaces and Linear Spaces. Then, starting with the chapter on Semi-norms, a general theory of Banach and Hilbert spaces is presented in connection with the theory of generalized functions of S. L. SOBOLEV and L. SCHWARTZ. While the book is primarily addressed to graduate students, it is hoped it might prove useful to research mathe- maticians, both pure and applied. The reader may pass, e. g. , from Chapter IX (Analytical Theory of Semi-groups) directly to Chapter XIII (Ergodic Theory and Diffusion Theory) and to Chapter XIV (Integration of the Equation of Evolution). Such materials as "Weak Topologies and Duality in Locally Convex Spaces" and "Nuclear Spaces" are presented in the form of the appendices to Chapter V and Chapter X, respectively. These might be skipped for the first reading by those who are interested rather in the application of linear operators.Lectures on the Calculus of Variations and Optimal Control Theory L. C. YoungThis book is divided into two parts. The first addresses the simpler variational problems in parametric and nonparametric form. The second covers extensions to optimal control theory. The author opens with the study of three classical problems whose solutions led to the theory of calculus of variations. They are the problem of geodesics, the brachistochrone, and the minimal surface of revolution. He gives a detailed discussion of the Hamilton-Jacobi theory, both in the parametric and nonparametric forms. This leads to the development of sufficiency theories describing properties of minimizing extremal arcs. Next, the author addresses existence theorems. He first develops Hilbert's basic existence theorem for parametric problems and studies some of its consequences. Finally, he develops the theory of generalized curves and ``automatic'' existence theorems. In the second part of the book, the author discusses optimal control problems. He notes that originally these problems were formulated as problems of Lagrange and Mayer in terms of differential constraints. In the control formulation, these constraints are expressed in a more convenient form in terms of control functions. After pointing out the new phenomenon that may arise, namely, the lack of controllability, the author develops the maximum principle and illustrates this principle by standard examples that show the switching phenomena that may occur. He extends the theory of geodesic coverings to optimal control problems. Finally, he extends the problem to generalized optimal control problems and obtains the corresponding existence theorems.Wigner Measure and Semiclassical Limits of Nonlinear Schrodinger Equations Ping Zhang###############################################################################################################################################################################################################################################################Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation William P. ZiemerThe major thrust of this book is the analysis of pointwise behavior of Sobolev functions of integer order and BV functions (functions whose partial derivatives are measures with finite total variation). The development of Sobolev functions includes an analysis of their continuity properties in terms of Lebesgue points, approximate continuity, and fine continuity as well as a discussion of their higher order regularity properties in terms of Lp-derivatives. This provides the foundation for further results such as a strong approximation theorem and the comparison of Lp and distributional derivatives. Also included is a treatment of Sobolev-Poincaré type inequalities which unifies virtually all inequalities of this type. Although the techniques required for the discussion of BV functions are completely different from those required for Sobolev functions, there are similarities between their developments such as a unifying treatment of Poincaré-type inequalities for BV functions. This book is intended for graduate students and researchers whose interests may include aspects of approximation theory, the calculus of variations, partial differential equations, potential theory and related areas. The only prerequisite is a standard graduate course in real analysis since almost all of the material is accessible through real variable techniques.Sphere Packings Chuanming Zong, John TalbotSphere packings is one of the most fascinating and challenging subjects in mathematics. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. This book gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and its proof methods. The book includes both classical and contemporary results and provides a full treatment of the subject. |